\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{5 x^{5}}+\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d \,x^{4}}-\frac {7 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{15 d^{2} x^{3}}-\frac {e^{5} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{4 d^{2}}+\frac {e^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4 d \,x^{2}} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^6/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{960} \, {\left (\frac {240 \, e^{4} \log \left (\sqrt {\frac {2 \, d}{x e + d} - 1} + 1\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{2}} - \frac {240 \, e^{4} \log \left ({\left | \sqrt {\frac {2 \, d}{x e + d} - 1} - 1 \right |}\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{2}} + \frac {8 \, {\left (15 \, e^{4} \log \left (2\right ) - 30 \, e^{4} \log \left (i + 1\right ) + 56 i \, e^{4}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{2}} - \frac {15 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 250 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 128 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 70 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 15 \, \sqrt {\frac {2 \, d}{x e + d} - 1} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{2} {\left (\frac {d}{x e + d} - 1\right )}^{5}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________